Foundations of Data Science

d-dimensional lattice except that the vertices in the same coordinate line form a clique.
To generate samples of x = (x 1 , . . . , x d ) with a target distribution p (x), the Gibbs
sampling algorithm repeats the following steps. One of the variables x i is chosen to be
updated. Its new value is chosen based on the marginal probability of x i with the other
variables fixed. There are two commonly used schemes to determine which x i to update.
One scheme is to choose x i randomly, the other is to choose x i by sequentially scanning
from x 1 to x d .
Suppose that x and y are two states that differ in only one coordinate. Without loss
of generality let that coordinate be the first. Then, in the scheme where a coordinate is
randomly chosen to modify, the probability p xy of going from x to y is
The normalizing constant is 1/d since ∑ y 1
d coordinates gives a value of d. Similarly,
p xy = 1 d p(y 1|x 2 , x 3 , . . . , x d ).
p(y 1 |x 2 , x 3 , . . . , x d ) equals 1 and summing over
p yx = 1 d p(x 1|y 2 , y 3 , . . . , y d )
= 1 d p(x 1|x 2 , x 3 , . . . , x d ).
Here use was made of the fact that for j ≠ 1, x j = y j .
It is simple to see that this chain has stationary probability proportional to p (x).
Rewrite p xy as
p xy = 1 p(y 1 |x 2 , x 3 , . . . , x d )p(x 2 , x 3 , . . . , x d )
d p(x 2 , x 3 , . . . , x d )
= 1 p(y 1 , x 2 , x 3 , . . . , x d )
d p(x 2 , x 3 , . . . , x d )
= 1 p(y)
d p(x 2 , x 3 , . . . , x d )
again using x j = y j for j ≠ 1. Similarly write
p yx = 1 p(x)
d p(x 2 , x 3 , . . . , x d )
from which it follows that p(x)p xy = p(y)p yx . By Lemma 5.3 the stationary probability
of the random walk is p(x).
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